Use the limit proces to find the are aof the region between the graph of the function f(x)=(-x^2)+4 and the x axis over the interval [0,2].
To find the area of the region between the graph of the function f(x) = (-x^2) + 4 and the x-axis over the interval [0,2], we can use the fundamental theorem of calculus. However, since you specifically mentioned using the limit process, let's proceed with that approach.
The area between the graph and the x-axis can be approximated using rectangles. We can start by dividing the interval [0,2] into n sub-intervals and then constructing rectangles that have widths equal to these sub-intervals. As n approaches infinity, the sum of the areas of these rectangles will approach the actual area we are trying to find.
To determine the height of each rectangle, we need to evaluate the function f(x) = (-x^2) + 4 at the right endpoint of each sub-interval within [0,2]. To do this, we need to find the right endpoint of each sub-interval, which can be calculated using the formula x = a + (i * Δx), where i is the index of the sub-interval (from 0 to n-1), a is the starting point of the interval (0 in this case), and Δx is the width of each sub-interval (Δx = (2-0)/n).
So, the right endpoint of the i-th sub-interval can be found by substituting the values into x = 0 + (i * (2/n)). Let's call this value x_i.
Now, we can calculate the height of each rectangle by evaluating f(x) = (-x^2) + 4 at x = x_i. So, the height of the i-th rectangle, h_i, can be found by substituting x_i into the function f(x) = (-x^2) + 4. Now we have the width (Δx) and the height (h_i) of each rectangle.
To calculate the area of each rectangle, we multiply the width and the height, resulting in A_i = Δx * h_i. Summing up the areas of all the rectangles, we have:
Total Area = Σ A_i
To approximate the actual area between the graph and the x-axis, we need to take the limit as n approaches infinity:
Actual Area = lim(n -> ∞) Σ A_i
This limit represents the integral of the function f(x) = (-x^2) + 4 over the interval [0,2]. By evaluating this limit, you can find the exact area of the given region.